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what is the definition of addition?

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The simple answer is that "adding" is basically "combining amounts."  Using the arabic notation we ultimately have to just remember certain things you can find an addition table here:

http://1.bp.blogspot.com/_pflilF-BfLQ/SwWx91O6R0I/AAAAAAAAABY/Ga8SI_ALc3A/s1600/addition_table+copy.jpg

If you look at the vertical column that has the numeral 6 at the very top in a pinkish box where the number itself is white, there are many numbers in white boxes below it with black text.  Now look at the row where there is a 4 all the way to the left in the pinkish box.  If you look at where the row and the column intersect you can see the number 10.  This means that 4 + 6 = 10.  If you look at the column that has the number 4 in the pink box and the row that has the number 6 in the pink box, you will find that the intersection is also 10.  This means that 6 + 4 = 4 + 6 = 10.

Basically if you start with 4 things, and then add 6 more things then you end up with 10 things.  Similarly if you started with 6 things and add 4 more you end up with 10 things still.  We say that addition is commutative for this reason.

There are lots more things that can be said about addition and the definition of addition can get pretty complicated if you get into the nitty gritty of mathematics.  If you are trying to add bigger numbers like 154 + 672 (which equals 826 by the way) that involves understanding what is meant by the number 72, which is actually a little complicated, so don't feel bad about asking this question.

I know that this might have been a bit more complicated than you were looking for so let me and other people on this site know if you need clarification, or have any other questions.